Treewidth Some problems parametrized by treewidth Results Further research The role of planarity in connectivity problems parameterized by treewidth Julien Baste and Ignasi Sau 1/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
Treewidth Some problems parametrized by treewidth Definition Results Courcelle Further research Tree decomposition Definition (Tree decomposition) Let G = ( V , E ) be a graph, a tree-decomposition of width w of G is a pair ( T , σ ), where T is a tree and σ = { B t | B t ⊆ V , t ∈ V ( T ) } such that : � t ∈ V ( T ) B t = V ( G ), For every edge { u , v } ∈ E ( G ) there is a t ∈ V ( T ) such that { u , v } ⊆ B t , B i ∩ B k ⊆ B j for all { i , j , k } ⊆ V ( T ) such that j lies on the path i , . . . , k in T , | B t | ≤ w + 1 for every t ∈ V ( T ). The set B t are called bag . 2/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
Treewidth Some problems parametrized by treewidth Definition Results Courcelle Further research Tree decomposition Definition (treewidth) The treewidth of G noted tw is the minimal value w such that there is a tree-decomposition of G of width w . An optimal tree decomposition is a tree-decomposition of width tw . [Roberson and Seymour.] 3/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
Treewidth Some problems parametrized by treewidth Definition Results Courcelle Further research Courcelle Theorem Theorem (Courcelle) Each graph property definable in monadic second-order logic can be decided in time f ( tw ) · n for some function f where n is the number of vertices of the input graph and tw its treewidth. [Courcelle] In this theorem the function f is a tower of exponential. 4/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? A simple problem Vertex Cover Input: A graph G = ( V , E ) and an integer k Question: Can we find X ⊆ V such that for all ( x , y ) ∈ E , x ∈ X or y ∈ X ? We have algorithm in time 2 O ( tw ) · n O (1). It is the best possible under ETH. [Impagliazzo, Paturi, Zane. JCSS 01] 5/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? Some more complicated problems Longest Path Input: A graph G = ( V , E ) and an interger k . Question: Does there exist a simple path of length k in G ? The best algorithm known in general graphs is in time 2 O ( tw log tw ) · n O (1) . 6/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? Some more complicated problems Longest Path Input: A graph G = ( V , E ) and an interger k . Question: Does there exist a simple path of length k in G ? The best algorithm known in general graphs is in time 2 O ( tw log tw ) · n O (1) . We have other examples of such problems : Steiner Tree Connected vertex cover ... 6/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? Some more complicated problems Longest Path Input: A graph G = ( V , E ) and an interger k . Question: Does there exist a simple path of length k in G ? The best algorithm known in general graphs is in time 2 O ( tw log tw ) · n O (1) . We have other examples of such problems : Steiner Tree Connected vertex cover ... These problems are called connectivity problems. 6/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? Meanwhile in sparse graphs Based on catalan structures, There are algorithms for these problems computing in time 2 O ( tw ) · n O (1) for some sparse graphs. In planar graphs. [Dorn, Penninkx, Bodlaender, and Fomin. ESA 2005] In graph of bounded genus. [Ru´ e, Sau, and Thilikos. ICALP 2010] H -minor free graph. [Dorn, Fomin, and Thilikos. SODA 2008] 7/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? New algorithms We recently have two new algorithms for many connectivity problems Cut & Count algorithm : [Cygan, Nederlof, Pilipczuk, Pilipczuk, Van Rooij and Wojtaszczyk. FOCS 11] Give randomize algorithm in time 2 O ( tw ) · n O (1) . Give connectivity problems that cannot be solved in time 2 o ( tw log tw ) · n O (1) 8/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? New algorithms We recently have two new algorithms for many connectivity problems Cut & Count algorithm : [Cygan, Nederlof, Pilipczuk, Pilipczuk, Van Rooij and Wojtaszczyk. FOCS 11] Give randomize algorithm in time 2 O ( tw ) · n O (1) . Give connectivity problems that cannot be solved in time 2 o ( tw log tw ) · n O (1) Rank based algorithm : [Bodlaender, Cygan, Kratsh, and Nederlof. ICALP 2013] Give deterministic in time 2 O ( tw ) · n O (1) . 8/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? What now? Type 1 Type 2 no 2 O ( tw ) · n O (1) 2 o ( tw log tw ) · n O (1) in general in general graphs graphs 9/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
A simple problem Treewidth Some more complicated problems Some problems parametrized by treewidth Sparse graphs Results Optimal algorithms Further research What now? What now? Type 1 Type 2 Type 3 no 2 o ( tw log tw ) · n O (1) no 2 o ( tw log tw ) · n O (1) 2 O ( tw ) · n O (1) in general graphs in general graphs in general 2 O ( tw ) · n O (1) no 2 o ( tw log tw ) · n O (1) graphs in planar graphs in planar graphs 10/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
Treewidth Type 1 Some problems parametrized by treewidth Type 2 Results Type 3 Further research Problem of type 1 Type 1 : Problems that can be solved in time 2 O ( tw ) · n O (1) on general graphs. 3-colorability Input: A graph G = ( V , E ). Question: Is there a color function c : V → { 1 , 2 , 3 } such that for all { x , y } ∈ E , c ( x ) � = c ( y ). Theorem Planar 3-colorability cannot be solved in time 2 o ( tw ) · n O (1) unless the ETH fails. 11/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
Treewidth Type 1 Some problems parametrized by treewidth Type 2 Results Type 3 Further research Problem of type 2 Type 2: No algorithm in time 2 o ( tw log tw ) · n O (1) on general graphs An algorithm in time 2 O ( tw ) · n O (1) when restricted to planar graphs. We find some problems of type 2: Cycle packing Max Cycle Cover Maximally Disconnected Dominating Set Maximally Disconnected Feedback Vertex Set 12/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
Treewidth Type 1 Some problems parametrized by treewidth Type 2 Results Type 3 Further research Problem of type 3 Type 3: An algorithm in time 2 O ( tw log tw ) · n O (1) on general graphs. No algorithm in time 2 o ( tw log tw ) · n O (1) even when restricted to planar graphs. Monochromatic Disjoint Paths A graph G = ( V , E ) of treewidth tw , a color function Input: γ : V → { 0 , . . . , tw } , m ∈ N and N = {N i = { s i , t i }| i ∈ { 1 , . . . , m } , s i , t i ∈ V } Question: Does G contain m pairwise vertex-disjoint monochro- matic paths from s i to t i , i ∈ { 1 , . . . , m } ? 13/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
Treewidth Type 1 Some problems parametrized by treewidth Type 2 Results Type 3 Further research k × k hitting set k × k hitting set Input: A family of sets S 1 , S 2 , . . . , S m ⊆ [ k ] × [ k ], such that each set contains at most one element from each row of [ k ] × [ k ] Question: Is there a set S containing exactly one element of each row such that S ∩ S i � = ∅ for any 1 ≤ i ≤ m ? [Lokshtanov, Marx, Saurabh. SODA 2011] Theorem k × k hitting set cannot be solved in time 2 o ( k log k ) unless the ETH fails. 14/22 Julien Baste and Ignasi Sau The role of planarity in connectivity problems parameterized
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